Do I have to collect everything?

First, i've played the Wii U version of Yoshi's Wooly World and as far as i know, the poochy version / DS version is mostly the same. So my answer is based on the WiiU version, but poochy owners feel free to correct me.

There are a three sets of collectibles each level that unlock the following things:

• Flowers: Collecting all flowers in all levels of a world unlocks a special bonus level that is harder than most levels present in the game. Collecting all flowers in the bonus levels unlocks a 'super' bonus level that is very long (and fun)
• Wool: Collecting all 5 wool bundles in a level unlocks a new color/pattern for yoshi, which you can alter on the main island
• Stamps: Colleting all 20 stamps in a level unlocks stamps that you can use in the MiiVerse. They have no effect on the game (other than 100% completion)

There is also an indicator for having all collectibles and 20/20 hearts at the end of the level, which is a (purely cosmetic) gold star. It makes you a bit cooler ofcourse :)

Sources: Own 100% playthrough, IGN, Prima Games